As noted by Turner and Widrow
[45]
inflation (for a comprehensive introduction to inflation see Ref.
[46])
provides four important ingredients for the production of primeval
magnetic fields.

Inflation naturally produces effects on very large scales,
larger than the Hubble horizon, starting from microphysical
processes operating on a causally connected volume. If
electromagnetic quantum fluctuations are amplified during
inflation they could appear today as large-scale static magnetic
fields (electric field should be screened by the high conductivity
plasma).

Inflation also provides the dynamical means to amplify these
long-wavelength waves. If the conformal invariance of the
electromagnetic field is broken in some way (see below) magnetic
fields could be excited during the de Sitter Universe expansion.
This phenomenon is analogous to particle production occurring in
a rapidly changing space-time metric.

During inflation (and perhaps during most of reheating) the Universe
is not a good conductor so that magnetic flux is not conserved and
the ratio r of the magnetic field with the radiation energy
densities can increase.

Classical fluctuations with wavelength
H-1 of massless, minimally coupled fields can grow
super-adiabatically, i.e. their energy density decrease only as
~ a-2 rather than a-4.

The main obstacle on the way of this nice scenario is given by the
fact that in a conformally flat metric, like the Robertson-Walker
usually considered, the background gravitational field does not
produce particles if the underlying theory is conformally
invariant [192].
This is the case for photons since the
classical electrodynamics is conformally invariant in the limit of
vanishing fermion masses. Several ways out this obstacle have been
proposed. Turner and Widrow
[45]
considered three
possibilities. The first is to break explicitly conformal
invariance by introducing a gravitational coupling, like RAµAµ or
RµAµAµ, where R is
the curvature scalar,
Rµ is
the Ricci tensor, and Aµ is the
electromagnetic field. These terms breaks gauge invariance and
give the photons an effective, time-dependent mass. In fact, one
of the most severe constraints to this scenario come from the
experimental upper limit to the photon mass, which today is
m < 2 × 10-16 eV
[193].
Turner and Widrow
showed that for some suitable (though theoretically unmotivated)
choice of the parameters, such a mechanism may give rise to
galactic magnetic fields even without invoking the galactic dynamo.
We leave to the reader to judge if such a
booty deserve the abandonment of the theoretical prejudice in
favor of gauge invariance. A different model invoking a
spontaneous breaking of gauge symmetry of electromagnetism,
implying nonconservation of the electric charge, in the early
stage of the evolution of the Universe has been proposed by Dolgov
and Silk [194].

The breaking of the conformal invariance may also be produced by
terms of the form
RµFµF /
m2 or RFµFµ,
where m is some mass scale required by dimensional
considerations. Such terms arise due to one-loop vacuum
polarization effects in curved space-time, and they have the
virtue of being gauge invariant. Unfortunately, Turner and Widrow
showed that they may account only for a far too small contribution
to primordial magnetic fields. The third way to break conformal invariance
discussed by Turner and Widrow invoke a coupling of the photon to
a charged field which is not conformally coupled or the anomalous
coupling to a pseudoscalar. This mechanism was already illustrated
in the previous section.

The anomaly can give rise to breaking of the conformal invariance
also in a different way. The kind of anomaly we are now discussing
about is the conformal anomaly, which is related to the triangle
diagram connecting two photons to a graviton. It is known (for a
review see Ref.
[195])
that this kind of diagrams breaks
conformal invariance by producing a nonvanishing trace of the
energy-momentum tensor

(4.77)

where is the
fine-structure constant of the theory based
on the SU(N) gauge-symmetry with Nf
fermion families, and

(4.78)

Dolgov [196]
pointed-out that such an effect may lead to
strong electromagnetic fields amplification during inflation.
In fact, Maxwell equations are modified by the anomaly in the
following way

(4.79)

which, in the Fourier space, gives rise to the equation

(4.80)

where A is the amplitude of the vector potential, and a prime
stands for a derivation respect to the conformal time
. At
the inflationary stage, when a' / a = - 1 /
Dolgov found a
solution of (4.80) growing like (H /
k)/2. Since k-1 grows well above the
Hubble radius during the de Sitter phase, a huge amplification can
be obtained if
>
0. Dolgov showed that for
~ 1
the magnetic field generated during the inflationary stage can be large
enough to give rise to the observed fields in galaxies even
without a dynamo amplification. Unfortunately, such a large value
of seems to
be unrealistic
( 0.06 for SU(5)
with three charged fermions). The conclusion is that galactic
magnetic fields might be produced by this mechanism only invoking
a group larger than SU(5) with a large number of fermion
families, and certainly no without the help of dynamo
amplification.

As we discussed in the above conformal invariance of the electromagnetic
field is generally
spoiled whenever the electromagnetic field is coupled to a scalar
field. Ratra [197]
suggested that a coupling of the form
eFµFµ,
where is a
arbitrary parameter, may lead to a huge amplification of
electromagnetic quantum fluctuations into large scale magnetic
fields during inflation. Such a coupling is produced in some
peculiar models of inflation with an exponential inflaton
potential [198].
It should be noted by the reader that the
scalar field coincide here
with the inflaton field. According to
Ratra, present time intergalactic magnetic fields as large as
10-9 G may be produced by this mechanism which would not
require any dynamo amplification to account for the observed
galactic fields. Unfortunately, depending on the parameter of the
underling model, the predicted field could also be as low as
10-65 G !

A slightly more predictive, and perhaps theoretically better
motivated, model has been proposed independently by Lemoine and Lemoine
[199] ,
and Gasperini, Giovannini and Veneziano
[200],
which is based on superstring cosmology
[201].
This model is based on the consideration that
in string theory the electromagnetic fields is coupled not only to
the metric
(gµ),
but also to the dilaton field
.
In the low energy limit of the theory, and after dimensional
reduction from 10 to 4 space-time dimensions, such a coupling
takes the form

(4.81)

which breaks conformal invariance of the electromagnetic field
and coincides with the coupling considered by Ratra
[197]
if = -1. Ratra, however,
assumed inflation to be driven by the scalar field potential,
which is not the case in string cosmology. In fact, typical
dilaton potentials are much too steep to produce the required
slow-roll of the inflaton (= dilaton) field. According to string
cosmologists, this problem can be solved by assuming inflation to
be driven by the kinetic part of the dilaton field, i.e. from
'
[201].
In such a scenario the Universe
evolves from a flat, cold, and weakly coupled
( =
- )
initial unstable vacuum state toward a curved, dilaton-driven,
strong coupling regime. During this period, called
pre-big-bang phase, the scale factor and of the dilaton evolve as

(4.82)

with > 1
and < 0. At
>
- 1 it begins
the standard FRW phase with a radiation dominated Universe. In the
presence of the non-trivial dilaton background the modified
Maxwell equations takes the form
[199]

(4.83)

Electromagnetic field amplification from quantum fluctuations
take place during the pre-big-bang phase when
' =
/
, where
=
. By following the
evolution of the electromagnetic field modes from t =
-
to now, Lemoine and Lemoine estimated that, in the most simple
model of dilaton-driven inflation (with
V() =
= p = 0)
a very tiny magnetic field is predicted today

(4.84)

where H1 = 1 /
1, which is far
too small to account for galactic fields.

Gasperini, Giovannini and Veneziano
[200],
reached a different conclusion, claiming that magnetic fields as large
those required to explain galactic fields without dynamo amplification
may be produced on the protogalactic scale. The
reason for such a different result is that they assumed a new
phase to exist between the dilaton-dominated phase and the FRW
phase during which dilaton potential is nonvanishing. The new
phase, called the string phase, should start when the string
length scale
S becomes
comparable to the horizon size at the conformal time
S
[202].
Unfortunately, the duration of such a phase is quite unknown, which
makes the model not very predictive.

Recently, several papers have been published (see e.g. Refs.
[203,
204,
205])
which proposed the generation of magnetic fields
by fluctuations of scalar (or pseudo-scalar) fields which were amplified
during, or at the end, of inflation. Typically, all these models
predict magnetic fields which need dynamo amplification to account for
the observed galactic and cluster fields.